If the net force acting on an object is zero, the velocity of that object does not change. \\
Conversely, if the velocity of an object doesn't change, the net force acting on it is zero.
\medskip
In the context of vectors, the ``net force'' is the sum of all the force vectors acting on the object. ``Speed'' is the length (or \textit{magnitude}) of the velocity vector.
\problem{}
There are no forces acting on the object
$A$ below. The current velocity of the object,
in meters per second, is represented by the vector $\overrightarrow{v}$.
Draw the position of the object two seconds later.
\begin{center}
\begin{normalsize}
\begin{tikzpicture} [scale=.7]
\draw[step=1cm, gray, very thin] (-11,-7) grid (1,1);
\draw [line width = 1.5pt, ->] (-9,0) -- (-5,-3);
\coordinate [label = above:{$\overrightarrow{v}$}] (v) at (-6.9,-1.5);
\filldraw (-9,0) circle (3pt);
\coordinate [label = left:{$A$}] (a) at (-9,0);
\end{tikzpicture}
\end{normalsize}
\end{center}
The sides of the grid squares
on the picture above are one metre long.
What is the speed of the object?
\vfill
What distance would the object cover
in two seconds? \\
\vfill
\pagebreak
\problem{}
There are no forces acting on the object
$A$ below.
The current velocity of this object,
in miles per hour,
is represented by the vector $\overrightarrow{v}$.
Draw the position of the object half an hour later.
\vspace{20pt}
\begin{center}
\begin{normalsize}
\begin{tikzpicture} [scale=.7]
\draw[step=1cm, gray, very thin] (-11,-7) grid (1,1);
\draw [line width = 1.5pt, ->] (-9,0) -- (-1,-6);
\coordinate [label = above:{$\overrightarrow{v}$}] (v) at (-4.8,-3);
\filldraw (-9,0) circle (3pt);
\coordinate [label = left:{$A$}] (a) at (-9,0);
\end{tikzpicture}
\end{normalsize}
\end{center}
The sides of the grid squares
on the picture above are ten miles long.
What is the speed of this object?
\vfill
What distance would the object cover
in half an hour? \\
\vfill
What distance would the object cover
in three hours? \\
\vfill
\pagebreak
\definition{Acceleration}
\textit{Acceleration} is the rate at which velocity changes.
Let's represent acceleration by the vector $\overrightarrow{a}$.
If $\overrightarrow{a}$ does not change over time, then the speed
of an object at time $t$ is given by the following equation:
\begin{equation}\label{eq:ac}
\overrightarrow{v_t} = \overrightarrow{v_0} +
t \overrightarrow{a}.
\end{equation}
\problem{}
It takes a minivan seven seconds to accelerate from 0 to 60 miles per hour. Find its acceleration in meters per second squared. \\
\hint{1 mile $\approx$ 1600 meters}
\vfill
Note that in the previous problem, motion is one-dimensional (it happens on a straight line). In this case, both velocity and acceleration are one-dimensional vectors---in other words, (real) numbers! \\
In general, velocity and acceleration are \textit{not} numbers, but vectors. You'll see this in the next few problems.
\pagebreak
Now, let's try a few examples with vectors: \\
\medskip
Consider an object, currently at $P_0$, moving along the vector $\overrightarrow{v_0}$.
As before, let $\overrightarrow{a}$ represent the acceleration of the object. This could be caused by gravity, current, or any other constant force.
\medskip
One second later, the object will be at $P_1$, and has the velocity vector $\overrightarrow{v_1}=\overrightarrow{v_0}+\overrightarrow{a}$.
\medskip
Two seconds later, the object will be positioned at $P_2$ and will have the velocity vector $\overrightarrow{v_2}=\overrightarrow{v_0}+2\overrightarrow{a}$.
\begin{center}
\begin{normalsize}
\begin{tikzpicture} [scale=.7]
\draw[step=1cm, gray, very thin] (-19,-15) grid (1,1);
Here, $\overrightarrow{F}$ is the net force acting on an object with mass $m$, and $\overrightarrow{a}$ is the acceleration the object experiences as a result of this action. Mass is a measure of an object's \textit{inertia}: the heavier an object is, the more effort it takes to change its velocity. \\
In civilized countries, mass is measured in grams and force is measured in \textit{newtons}. One newton is the force it takes to accelerate 1 kg of mass to 1 meter per second. In other words,
In other words, this means that when one object exerts force on another, the second simultaneously exerts a force equal in magnitude and opposite in direction to the force exerted on it by the first.
Here is an important example of an inverse vector. When you stand still, the floor pushes you up with the force opposite to the force of the gravitational pull, a.k.a. \textit{weight}.
